A002595 - OEIS
1, 6, 40, 112, 1152, 2816, 13312, 10240, 557056, 1245184, 5505024, 12058624, 104857600, 226492416, 973078528, 2080374784, 23622320128, 30064771072, 635655159808, 446676598784, 11269994184704, 23639499997184, 6597069766656
COMMENTS
arcsin(x) is usually written as x + x^3/(2*3) + 1*3*x^5/(2*4*5) + 1*3*5*x^7/(2*4*6*7) + ..., = x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ... when reduced to lowest terms.
arccos(x) = Pi/2 - (x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ...).
arccsc(x) = 1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+...
arcsec(x) = Pi/2 -(1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+...)
arcsinh(x) = x-1/6*x^3+3/40*x^5-5/112*x^7+35/1152*x^9-63/2816*x^11+...
arccsc(x) = arcsin(1/x) and arcsec(x) = arccos(1/x): 1 < |x|
arccsch(x) = arcsinh(1/x) for 1 < |x|
Also denominator of (2n-1)!! / ((2n+1)*(2n)!!) (n=>0).
REFERENCES
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, Chap. 3.
Focus, vol. 16, no. 5, page 32, Oct 1996.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 31, equation 31:6:1 at page 290.
FORMULA
a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))). - Johannes W. Meijer, Jul 06 2009
MATHEMATICA
Denominator[Take[CoefficientList[Series[ArcSin[x], {x, 0, 50}], x], {2, -1, 2}]] (* Harvey P. Dale, Aug 06 2012 *)
PROG
(PARI) a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))); \\ Stefano Spezia, Dec 31 2024
KEYWORD
nonn,frac,nice,easy,changed